There are 24*60 minutes in a day (ignoring the imperfections of the natural world, the Earth and Sun). So there are 24*60 valid 24 hour times (excluding seconds) on a digital clock.

Each of these can be rotated 4 ways, by 0,1,2 or 3 places. For example :

1 - 12:34

2 - 23:41

3 - 34:12

4 -4 1:23

In which case only 1 and 2 are valid 24 hour times.

How many of these 4*24*60 rotations are also valid minutes? (What is the fastest way to find this out?)